A372001 Array read by descending antidiagonals: A family of generalized Catalan numbers generated by a generalization of Deléham's Delta operator.

1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 14, 15, 5, 1, 1, 42, 105, 61, 9, 1, 1, 132, 945, 1385, 297, 17, 1, 1, 429, 10395, 50521, 24273, 1585, 33, 1, 1, 1430, 135135, 2702765, 3976209, 485729, 8865, 65, 1, 1, 4862, 2027025, 199360981, 1145032281, 372281761, 10401345, 50881, 129, 1, 1

Offset: 1

Author

Peter Luschny, Apr 21 2024

Keywords

• nonn
• tabl

Comments

Deléham's Delta operator is defined in A084938. It maps two sequences (a, b) to a triangle T. The given sequences are the coefficients of the linear function p = a + x*b which is the starting point of a recurrence described in A084938 and implemented in A371637. The generalization given here extends the number of input sequences to any number, mapping (a, b, c, ...) to p = a + x*b + x^2*c ... but leaves the recurrence unchanged.

The result, as said, is a triangle that we can evaluate in two ways: Firstly, we only return the main diagonal. In this case, we created a new sequence from n given sequences. This case is implemented by the function A(n, dim) below.

Alternatively, we return the entire triangle. But since the triangle is irregular, we convert it into a regular one by taking only every n-th term of a row. This case is handled by the function T(n, dim). For the first few triangles generated this way, see the link section.

Examples

			Array starts:
  [0] 1, 1,  2,     5,        14,            42,                132, ...
  [1] 1, 1,  3,    15,       105,           945,              10395, ...
  [2] 1, 1,  5,    61,      1385,         50521,            2702765, ...
  [3] 1, 1,  9,   297,     24273,       3976209,         1145032281, ...
  [4] 1, 1, 17,  1585,    485729,     372281761,       601378506737, ...
  [5] 1, 1, 33,  8865,  10401345,   38103228225,    352780110115425, ...
  [6] 1, 1, 65, 50881, 231455105, 4104215813761, 220579355255364545, ...
.
Seen as a triangle T(n, k) = A(k, n - k):
  [0] [  1]
  [1] [  1,     1]
  [2] [  2,     1,     1]
  [3] [  5,     3,     1,     1]
  [4] [ 14,    15,     5,     1,    1]
  [5] [ 42,   105,    61,     9,    1,  1]
  [6] [132,   945,  1385,   297,   17,  1, 1]
  [7] [429, 10395, 50521, 24273, 1585, 33, 1, 1]
		

Links

• Peter Luschny, First few triangles generated by A372001.

Crossrefs

By ascending antidiagonals: A290569.

Family: A000108 (n=0), A001147 (n=1), A000364 (n=2), A216966 (n=3), A227887 (n=4), A337807 (n=5), A337808 (n=6), A337809 (n=7).

Cf. A291333 (main diagonal), A371999 (row sums of triangle).

Cf. A084938, A131427, A371994, A371637.

Programs

SageMath

def GeneralizedDelehamDelta(F, dim, seq=True):  # The algorithm.
    ring = PolynomialRing(ZZ, 'x')
    x = ring.gen()
    A = [sum(F[j](k) * x^j for j in range(len(F))) for k in range(dim)]
    C = [ring(0)] + [ring(1) for i in range(dim)]
    for k in range(dim):
        for n in range(k, 0, -1):
            C[n] = C[n-1] + C[n+1] * A[n-1]
        yield list(C[1])[-1] if seq else list(C[1])
def F(n):  # Define the input functions.
    def p0(): return lambda n: pow(n, n^0)
    def p(k): return lambda n: pow(n + 1, k)
    return [p0()] + [p(k) for k in range(n + 1)]
def A(n, dim): # Return only the main diagonal of the triangle.
    return [r for r in GeneralizedDelehamDelta(F(n), dim)]
for n in range(7): print(A(n, 7))
def T(n, dim): # Return the regularized triangle.
    R = GeneralizedDelehamDelta(F(n), dim, False)
    return [[r[k] for k in range(0, len(r), n + 1)] for r in R]
for n in range(0, 4):
    for row in T(n, 6): print(row)

Formula

A = DELTA([x -> (x + 1)^k : 0 <= k <= n]), i.e. here the input functions of the generalized Delta operator are the (shifted) power functions. The returned sequence is the main diagonal of the generated triangle.